A reliable bus transfer collects you from the Hilton NBO and drives you and luggage thru Namanga to Arusha.
Only takes a couple hours and no need whatever to have a private driver, although I can't see what his problem would be to do this.
Get Tanzania Visa at Namanga - faster and cheaper than outside the country. We did that a couple months back.
There is no comparison between Kenya and Tanzania re game. I have seen both.
Kenya parks are largely ruined as they have had no controls on them for decades. Govt has allowed short term bucks ruin the pace for everyone. Actually the national park out near NBO airport will show you more game than the parks near Kili. Strange but true.
Tanzania is an object lesson on how to manage wildlife carefully and sustainably.
NgoroNgoro crater you will not see in Kenya.
Carnivore Restaurant you will not see in Tanzania - a MUST.
All you can eat Ostrich, and Kudu, crocodile, impala, eland, giraffe, etc lunch on skewers right from the coals for about $US20 a head. Every city tour type deal of NBO adds this for even less than $US20 and that included transfers.
Everyone in the area should add Zanzibar for a few days at either end. You can fly there direct from Arusha cheaply.